The subject matter described herein generally relates to a method and apparatus for determining sensitivities of reflection or transmission spectra to various grating parameters via a time or frequency domain simulation. The computation of these sensitivities may be central in a number of techniques used in optically based metrology tools to deduce the features on semiconductor wafers.
Some metrology tools measure more than just one-dimensional critical dimension (CD) features, because of smaller device dimensions and tighter process control windows. It has become critical to efficiently detect, identify, and measure changes in feature profiles to control current and future semiconductor lithography and etch processes.
Generally, the determination of sensitivities of reflection or transmission spectra to various grating parameters via a time or frequency domain simulation has used a finite difference approximation of the derivatives.
The method of computing derivatives by finite differences has a number of disadvantages. For example, each desired derivative with respect to a single scalar parameter requires an additional solution of Maxwell's Equation. This computation can be costly, and as a result limits the total number of derivatives that may be computed for application where computation time is important. The computation cost also limits the number of wavelengths from the optical scattering data that may be processed, which prevents the use of the full information provided by the optical scattering data.